On functional central limit theorems for dependent, heterogeneous arrays with applications to tail index and tail dependence estimation

We establish invariance principles for a large class of dependent, heterogeneous arrays. The theory equally covers conventional arrays, and inherently degenerate tail arrays popularly encountered in the extreme value theory literature including sample means and covariances of tail events and exceedances. For tail arrays we trim dependence assumptions down to a minimum leaving non-extremes and joint distributions unrestricted, covering geometrically ergodic, mixing, and mixingale processes, in particular linear and nonlinear distributed lags with long or short memory, linear and nonlinear GARCH, and stochastic volatility.Of practical importance the limit theory can be used to characterize the functional limit distributions of a tail index estimator, the tail quantile process, and a bivariate extremal dependence estimator under substantially general conditions.